# Questions on Kelly criterion

I am new to asset allocation problems and have some concerns regarding the derivation of the continuous-time Kelly criterion (i.e. not the original version destined to discrete sports betting/Casino).

I am following the derivation of Martin & Schöneborn, see page 4 and following.

As far as I understand it, the assumptions are:

1. Economy with 2 investment vehicles: a risky asset $X$ and a risk-free one $B$
2. $X$ is driven by a time-homogeneous diffusion process: $$dX_t = \mu(X_t) dt + \sigma (X_t) dW_t$$ while $B$ follows the usual: $dB_t = rB_t dt$
3. The investment strategy consists in holding $\{\theta_t\}_{t\geq 0}$ units of the risk-asset at any time $t \geq 0$. It is assumed self-financing, so that the P&L over an infinitesimal time interval $[t,t+dt[$ writes: $$d\Pi_t = \theta_t dX_t + dB_t$$
4. The investor aims at maximizing the total discounted expectation of the utility of her $\theta$-controlled P&L over a forward-looking horizon stretching from today $t$ up to infinity: $$V_t = \int_{s=t}^\infty e^{-r(s-t)} \mathbb{E}_t\left[ \mathcal{U}(\theta_s dX_s) \right]$$
5. Since $X_t$ is a diffusive process with bounded quadratic variation, a second order definition of the utility function $\mathcal{U}(.)$ is enough. Typically, we take: $$\mathcal{U}(0)=0,\ \ \mathcal{U}'(0)=1,\ \ \mathcal{U}''(0)=-1/G$$
6. Continuous trading is possible and there are no market frictions.

And the result is: $$\theta_t = \frac{\mu(X_t)}{\sigma^2(X_t)}G$$

What I don't understand is:

1. In the paper, they look for $\{\theta_t\}$ which maximises $V_t$. This leads to the following ODE $$\frac{\partial f}{\partial x}(X_t) \mu(X_t) + \frac{1}{2} \frac{\partial^2 f}{\partial x^2}(X_t) \sigma^2(X_t) - rf(X_t) = -\dot{U}(X_t, \theta_t)$$ Now they jump to the "intuitively clear conclusion" that this is equivalent to finding $\theta_t$ which maximises $\dot{U}(X_t, \theta_t)$. Unfortunately, it is not clear to me.
2. The fact that the optimal allocation strategy does not involve the risk-free rate $r$ seems weird to me. In my opinion, this comes from the fact that we should rather define the value function $V_t$ to be maximised as: $$V_t = \int_{s=t}^\infty e^{-r(s-t)} \mathbb{E}_t\left[ \mathcal{U}(\theta_s dX_s + dB_s) \right]$$
3. How do you use this Kelly allocation criterion in practice? I mean, suppose time $t$ corresponds to the market close of day $D$. It seems natural to want to know the optimal allocation (or position) one should take at $t+\Delta t$ (e.g. day $D+1$). So what we need is something like: $$\theta_{t+\Delta t} = f(\text{information contained in the filtration } \mathcal{F}_t)$$ and not $\theta_t$ on the LHS, since it is already known at time $t$ (in the demonstration we always assume $\theta_t$ to be adapted, hence measurable at any $t$). Should I assume a particular continuity assumption for $\theta_t$?
4. I've read somewhere that $\forall G$ the Kelly strategy is optimal in the following sense: $$\mathbb{E} \left[ \frac{ \Pi_T(\{\theta_t^{\text{Kelly}(G)}\}_{t\in[0,T]}) }{ \Pi_T( \{\theta_t\}_{t\in[0,T]}) } \right] \geq 1$$ for all $\{\theta_t\}_{t\in[0,T]}$ adapted to the natural filtration of $W_t$ and where $\Pi_T( \{\theta_t\}_{t\in[0,T]}$ denotes the final wealth of an investment strategy based on the risky asset position sizing $\{\theta_t\}$. But some simulations show me that this is not true, especially not $\forall G$. Even the growth rate $\dot{U}(\theta_t,X_t)$ mentioned in the paper does not seem optimal (sometimes I have basic long only or short only strategy that perform better).

Could someone please help me out? If the answer to all these questions is that this derivation is wrong and not the one I should follow, I would be glad if someone could provide a nice reference.

Cheers

• I don't know for the rest but concerning your second point, remember that all quantities are expressed in differential form with respect to the Itô integral definition. Notably $\theta_t dX_t$ simply means that $I_T = \int_0^T \theta_t dX_t$ is an Itô integral, where by definition $dX_t$ is forward-looking ($dX_t = X_{t+dt}-X_t$) and $\theta_t$ prevails over the interval $[t, t+dt[$. So basically, I would say that $\lim_{\epsilon \rightarrow 0} \theta_{t+dt-\epsilon} = \theta_t$. – Quantuple Jul 15 '16 at 14:05
• I applied Kelly crierion on sportsbook betting as well, you might try to browse over Application of Kelly Criterion model in Sportsbook Investment. Creating a Profitable Betting Strategy for Football by Using Statistical Modelling by Niko Marttinen (2006) might useful for you. – RYO ENG Lian Hu Sep 6 '17 at 3:46
• If I follow correctly, the "intuitively clear conclusion" is based on the premise that solving for zeros of a derivative also solves for the extrema of the original function. – David Addison Jan 13 '19 at 23:28